推 t0444564 :已收錄 07/12 10:19
※ 編輯: d3osef (111.243.102.90), 07/12/2014 13:52:23
課程名稱︰高等統計推論二
課程性質︰選修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰102.04.19
考試時限（分鐘）：110
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Advanced Statistical Inference II Midterm Examination 19 Apr 2014
1. A company wants to estimate the proportioin p, 0＜p＜1, of dedective items
it produces. Since they rarely produce defective items, n workers are asked
to continue inspecting until they each has observed one defective item. For
th
i=1,2,....,n, let X be the number of items the i worker inspect before
s/he has observed one defective item.
(a)5% What assumptions are needed so that the model X1,......,Xn i.i.d.
distributed as Geometric(p) is appropriate for this setting?
^ n ^
(b)10% Let p1= (n-1)(Σ Xi-1)^(-1). Show that p1 is the uniformly minimum
i=1
variance unbiased estimator of p.
(c)10% ^
Show that the maximum likelihood estimator of p, denoted as p2, is biased.
2. Suppose that X1,X2,....,Xn is an i.i.d sample from the Uniform(-θ,θ)
distribution which has density f(x)=2^(-1)θ^(-1)I (x), where I is
(-θ,θ)
the indicator funtion.
^
(a)8% Show that the maximum likelihood estimator of θ, denoted as θ1 is ^
biased and has a simple rescaling which is unbiased. Call this rescaling θ2.
^
(b)8% Is θ2 an uniformly minimum variance unbiased estimator of θ ?
Why or why not?
3. A random sample X=(X1,....,Xn) is drawn from a Pareto population with pdf
θυ^θ
f(x│θ,υ)= --------------I (x), θ＞0, υ＞0, where θ and υ are
x^(θ+1) [υ,∞)
both unknown.
(a)5% Show that the LRT of H0:θ=1 versus H1:θ≠1 has critical region of the
form:{x:T(x)≦c1 or T(x)≧c2} where 0＜c1＜c2 and
ΠXi
T=log [-----------].
min(Xi)^n
(b)10% Show that, under H0, 2T has a chi-square distribution and find the
number of degrees of freedom.
4. Let X1,X2,....Xi be iid Beta(μ,1) and Y1,Y2,....Yi be iid Beta(θ,1).
(a)10% Show that a LRT test of H0:θ=μ versusθ≠μ can be based on the
statistic ΣlogXi
T=---------------------
ΣlogXi + ΣlogYi
(b)10% Find the distribution of T when H0 is true, and then show how to get a
test of size α.
5. Let X be a single observation from the pdf f(x)=θx^(θ-1) I (x), where
(0,1)
θ＞0 is unknown.
(a)8% Let T= X^θ. Use T as a pivot construct 1-α confidence intervals for θ.
(b)8% Among the confidence intervals in (a), find the one that has the minimum
length.
(c)8% Find the 1-α confidence interval for θ by inverting the acceptance
regions of the LRTs H0: θ=θ0 versus H0:θ≠θ0.
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